Phases of Gauge Theories

July 2010 @ Quark Matter - Roma

Francesco Sannino

Cosmology

Atoms4%

Dark Matter22%

Dark Energy74%

Cosmology

Low Energy Effective TheoryStandard Model

Low Energy Effective Theory

SM

Standard Model

WW Scattering

WW Scattering

S-wave amplitude:

WW Scattering

S-wave amplitude:

WW Scattering

S-wave amplitude:

WW Scattering

S-wave amplitude:

Unitarity:

WW Scattering

Theorem:

Unitarity requires the existence of a weakly coupled Higgs particle or New Physics around the Terascale!

Origin of mass

The Higgs

Custodial symmetries

Custodial symmetries

Custodial symmetries

Custodial symmetries

SUL(2)× UY (1)

SUL(2)× UY (1)

m2 < 0

|H|

m2 < 0

�σ2� ≡ v2 = |m2|λ

|H|

m2 < 0

�σ2� ≡ v2 = |m2|λ

σ = v + h

Gauge Boson-Masses

Gauge Boson-Masses

Gauge Boson-Masses

Gauge Boson-Masses

−λd QL · HdR

Gauge Boson-Masses

Quark-Masses

−λd QL · HdR

Gauge Boson-Masses

Quark-Masses

−λd QL · HdR

Gauge Boson-Masses

Quark-Masses

SM Higgs: Current Status:

0

1

2

3

4

5

6

10030 300

mH [GeV]

!"2

Excluded Preliminary

!#had =!#(5)

0.02758±0.000350.02749±0.00012incl. low Q2 data

Theory uncertaintymLimit = 144 GeV

Higgs mechanism in Nature

Superconductivity

Macroscopic-ScreeningNon-Relativistic

SM-ScreeningRelativistic

Superconductivity

Macroscopic-ScreeningNon-Relativistic

SM-ScreeningRelativistic

Superconductivity

Macroscopic-ScreeningNon-Relativistic

SM-ScreeningRelativistic

Superconductivity

|H|2 =v2

2

Macroscopic-ScreeningNon-Relativistic

SM-ScreeningRelativistic

Superconductivity

Meissner-MassStatic Vector Potential

Weak-GB-Mass

Meissner-MassStatic Vector Potential

Weak-GB-Mass

Meissner-MassStatic Vector Potential

Weak-GB-Mass

Meissner-MassStatic Vector Potential

Weak-GB-Mass

Meissner-MassStatic Vector Potential

Weak-GB-Mass

Meissner-MassStatic Vector Potential

Weak-GB-Mass

Hidden structure

Meissner-MassStatic Vector Potential

Weak-GB-Mass

Hidden structure ????

Instability of the Fermi Scale

M2H

= 2λ v2

Quantum corrections

Quantum corrections

A mass appears even if ab initio is set to zero!

Quantum corrections

No custodial symmetry protecting a scalar mass.

A mass appears even if ab initio is set to zero!

Quantum corrections

Hierarchy between the EW scale and the Planck Scale.

No custodial symmetry protecting a scalar mass.

A mass appears even if ab initio is set to zero!

Quantum corrections

Definition of Natural

Small parameters stay small under radiative corrections.

Definition of Natural

Electron mass

If set to zero the U(1)L× U(1)R forbids its regeneration

Electron mass

If set to zero the U(1)L× U(1)R forbids its regeneration

Naturalness begs an explanation of the origin of mass.

Electron mass

If set to zero the U(1)L× U(1)R forbids its regeneration

Naturalness begs an explanation of the origin of mass.

No conflict with any small value of the electron mass

Electron mass

Low Energy Effective TheoryMany Models

(?)MSSMXLMS D

AdS/?Unparticle

Branes......

Technicolor

Low Energy Effective TheoryMany Models

(?)MSSMXLMS D

AdS/?Unparticle

Branes......

Technicolor

$

Theory landscape

Gauge Group

Theory landscape

Gauge Group

Matter

Theory landscape

Gauge Group

SUSY

Matter

Theory landscape

Gauge Group

SUSY

Matter

N=1

N=2

N=4

Theory landscape

Gauge Group

SUSY Non SUSY

Matter

N=1

N=2

N=4

Theory landscape

Gauge Group

SUSY Non SUSY

Matter

N=1

N=2

N=4

Fermions

Fermions + Bosons

Bosons

Theory landscape

Gauge Group

SUSY Non SUSY

Matter

N=1

N=2

N=4

Fermions

Fermions + Bosons

Bosons

Vector

Theory landscape

Gauge Group

SUSY Non SUSY

Matter

N=1

N=2

N=4

Fermions

Fermions + Bosons

Bosons

Vector Chiral

Theory landscape

Gauge Group

SUSY Non SUSY

Matter

N=1

N=2

N=4

Fermions

Fermions + Bosons

Bosons

Vector Chiral

Theory landscape

Gauge Group

SUSY Non SUSY

Matter

N=1

N=2

N=4

Fermions

Fermions + Bosons

Bosons

Vector Chiral

Theory landscape

Gauge Group

SUSY Non SUSY

Matter

N=1

N=2

N=4

Fermions

Fermions + Bosons

Bosons

Vector Chiral

Theory landscape

Phases

α(r)→ 1log(r)

UVUVIR IR

β =d g

d lnµα =g2

4π

V (r) ∝ 1r log(r)

Free Electric

UVUV IRIR

IR Conformal Phase

IR conformal behavior

V (r) ∝ 1r

Coulomb

UV

UV

IR

IR

V ∝ σ r

QCD - Like

Knobs

Gauge Group, i.e. SU, SO, SP

Knobs

Gauge Group, i.e. SU, SO, SP

Matter Representation

Knobs

Gauge Group, i.e. SU, SO, SP

Matter Representation

# of Flavors per Representation

Knobs

Gauge Group, i.e. SU, SO, SP

Matter Representation

# of Flavors per Representation

Knobs

Nf

Gauge Group, i.e. SU, SO, SP

Matter Representation

# of Flavors per Representation

Knobs

NfQCD

Gauge Group, i.e. SU, SO, SP

Matter Representation

# of Flavors per Representation

Knobs

NfQCD IR Conformal

Gauge Group, i.e. SU, SO, SP

Matter Representation

# of Flavors per Representation

Knobs

NfQCD IR Conformal NA-QED

Gauge Group, i.e. SU, SO, SP

Matter Representation

# of Flavors per Representation

Knobs

NfQCD IR Conformal NA-QED

SU(N)

Adjoint Dirac Matter

Nf

Matter Density (i.e. Chemical Potential(s)) =0

Temperature = 0

Example

SU(N)

Adjoint Dirac Matter

Nf

Matter Density (i.e. Chemical Potential(s)) =0

Temperature = 0

Example

SU(N)

Adjoint Dirac Matter

Nf

Matter Density (i.e. Chemical Potential(s)) =0

Temperature = 0

Example

SU(N)

Adjoint Dirac Matter

Nf

Matter Density (i.e. Chemical Potential(s)) =0

Temperature = 0

Example

SU(N)

Adjoint Dirac Matter

Nf

Matter Density (i.e. Chemical Potential(s)) =0

Temperature = 0

Example

β(g) = −β0

(4π)2g3−

β1

(4π)4g5 +O(g7)

β0 = 0

β(g) = −β0

(4π)2g3−

β1

(4π)4g5 +O(g7)

β0 = 0

β = 0

β(g) = −β0

(4π)2g3−

β1

(4π)4g5 +O(g7)

β0 = 0

β = 0

Weakly Coupled

β(g) = −β0

(4π)2g3−

β1

(4π)4g5 +O(g7)

β0 = 0

β = 0

Weakly Coupled

Strongly Coupled

β(g) = −β0

(4π)2g3−

β1

(4π)4g5 +O(g7)

β0 = 0

β = 0

Weakly Coupled

Strongly Coupled

Analytic Tools

The all orders beta function of NSVZ

Unitarity Bounds for Conformal Theories

Non-Renormalization of Superpotentials

‘t Hoofts Anomaly Matching Conditions

a-Maximization

Instanton Calculus

....

Seiberg, Intriligator, Novikov, Shifman, Vainshtein, Zakharov.

SUSY

Different approaches

Schwinger - Dyson

Instanton Inspired Calculus

Thermal degrees of freedom count

Exact Renormalization Methods

Certain Topological excitations

Appelquist, Bowick, Chivukula, Cohen, Eichten, Gies, Hill, Holdom, Karabali, Jaeckel, Fisher, Lane, Litim, Mahanta, Miransky, Pawlowski, Percacci, Poppitz, Shrock, Simmons, Terning, Unsal, Wijewardhana, Yamawaki

Non SUSY

Also:

The all orders beta function conjecture* (RS)

Chiral Gauge Theories beta function (F.S.)

Unitarity of the Operators for Conformal Theories

Gauge Dualities (F.S.)

First Principle Lattice Computations

RS - Ryttov, F.S. 07F.S. 09AT - Antipin, Tuominen 09

*Variations on the beta function (AT)

..continued

The full nonperturbative fermion propagator reads:

Schwinger - Dyson

The Euclidianized gap equation in Landau gauge is:

αc =π

3C2(r)

In practice

β(g) = − β0

(4π)2g3 − β1

(4π)4g5

β = 0

β(g) = − β0

(4π)2g3 − β1

(4π)4g5

β = 0

β(g) = − β0

(4π)2g3 − β1

(4π)4g5

β = 0α∗

4π= −β0

β1

β(g) = − β0

(4π)2g3 − β1

(4π)4g5

β = 0α∗

4π= −β0

β1

αc =π

3C2(r)

β(g) = − β0

(4π)2g3 − β1

(4π)4g5

β = 0α∗

4π= −β0

β1

αc =π

3C2(r)

β(g) = − β0

(4π)2g3 − β1

(4π)4g5

β = 0α∗

4π= −β0

β1

αc =π

3C2(r)γ(2− γ) = 1

β(g) = − β0

(4π)2g3 − β1

(4π)4g5

β = 0α∗

4π= −β0

β1

αc =π

3C2(r)γ(2− γ) = 1

β(g) = − β0

(4π)2g3 − β1

(4π)4g5

α∗ ≤ αc

N cf Ladder

=17C2(G) + 66C2(r)10C2(G) + 30C2(r)

C2(G)T (r)

C2(G) = C2(Adj) = N

SU(N) Dirac Fermions in representation “r”

SU(N) Adjoint Dirac Matter

Back to the example

SU(N) Adjoint Dirac Matter

C2(r) = C2(G) = T (G) = N

Back to the example

SU(N) Adjoint Dirac Matter

C2(r) = C2(G) = T (G) = N

β0 =113

N − 43NNf = 0 , → NAsymp

f =114

= 2.75

Back to the example

SU(N) Adjoint Dirac Matter

C2(r) = C2(G) = T (G) = N

N cf Ladder

=17 + 6610 + 30

= 2.075

β0 =113

N − 43NNf = 0 , → NAsymp

f =114

= 2.75

Back to the example

SU(N) Adjoint Dirac Matter

C2(r) = C2(G) = T (G) = N

N cf Ladder

=17 + 6610 + 30

= 2.075

β0 =113

N − 43NNf = 0 , → NAsymp

f =114

= 2.75

F.S. - Tuominen 04

Back to the example

SU(N) Adjoint Dirac Matter

NAsympf = 2.75

N cf Ladder

= 2.075

F.S. - Tuominen 04

Back to the example

Another Example

SU(N) 2-index Symmetric Rep

Diagram for Symm. Rep.

Diagram for Symm. Rep.

Dietrich and F.S. 06

Fund

2A

2S

Adj

Ladder approximation

SD - Phase Diagram

SUSY

SUSY β function

NSVZ

SUSY β function

NSVZ

SUSY β function

NSVZ

SUSY β function

NSVZ

SUSY β function

Ryttov and F.S. 07

Fund

2A

2SAdj

NSVZ, SeibergIntriligator-Seiberg

SUSY Phase Diagram

Ryttov and F.S. 07

β - function

β function conjecture

Ryttov and F.S. 07

β(g) = − g3

(4π)2β0 − 2

3T (r)Nfγ(g2)

1− g2

8π2 C2(G)(1 + 2β�0

β0)

β function conjecture

Ryttov and F.S. 07

γ = −d lnm/d lnµ

β(g) = − g3

(4π)2β0 − 2

3T (r)Nfγ(g2)

1− g2

8π2 C2(G)(1 + 2β�0

β0)

β function conjecture

Ryttov and F.S. 07

γ(g2) =32C2(r)

g2

4π2+ O(g4)γ = −d lnm/d lnµ

β(g) = − g3

(4π)2β0 − 2

3T (r)Nfγ(g2)

1− g2

8π2 C2(G)(1 + 2β�0

β0)

β function conjecture

Ryttov and F.S. 07

γ(g2) =32C2(r)

g2

4π2+ O(g4)γ = −d lnm/d lnµ

β0 =113

C2(G)− 43T (r)Nf

β�0 = C2(G)− T (r)Nf

β(g) = − g3

(4π)2β0 − 2

3T (r)Nfγ(g2)

1− g2

8π2 C2(G)(1 + 2β�0

β0)

β function conjecture

SU(N) Nf =12

Adjoint Matter

Recovering SYM

Ryttov and F.S. 07

SU(N) Nf =12

Adjoint Matter

Recovering SYM

Ryttov and F.S. 07

SU(N) Nf =12

Adjoint Matter

Recovering SYM

Ryttov and F.S. 07

SU(N) Nf =12

Adjoint Matter

Recovering SYM

Ryttov and F.S. 07

SU(N) Nf =12

Adjoint Matter

Recovering SYM

1.00.5 2.00.2 5.0 10.0

4

6

8

10

!!Gev"

g2N

All Orders2-loops1-loop

SU(2)SU(3)SU(4)

Luscher, Sommer, Wolff, Weisz, 92Luscher, Sommer, Weisz, Wolff 94Lucini and Moraitis 07

SU(2)

SU(3)SU(4)

Confronting with YM

Ryttov and F.S. 07

β = 0

Unitarity of the Conformal Operators demands:

γ ≤ 2

Bounding the window

Back to the Example

SU(N) Adjoint Dirac Matter

γ ≤ 2 , =⇒ N cf ≥

118

Confronting with YM

Back to the Example

SU(N) Adjoint Dirac Matter

γ ≤ 2 , =⇒ N cf ≥

118

γ = 1 , =⇒ N cf =

116

= 1.83

Confronting with YM

Back to the Example

SU(N) Adjoint Dirac Matter

γ ≤ 2 , =⇒ N cf ≥

118

N cf Ladder

= 2.075

γ = 1 , =⇒ N cf =

116

= 1.83

Confronting with YM

Back to the Example

SU(N) Adjoint Dirac Matter

γ ≤ 2 , =⇒ N cf ≥

118

N cf Ladder

= 2.075

γ = 1 , =⇒ N cf =

116

= 1.83

Conformal in the IR

Confronting with YM

SU(N) Adjoint Dirac Matter

NAsympf = 2.75

γ ≤ 2→ N cf ≥

118

Back to the example

Universal Picture

Fund

2A

2S

Adj

Ladder

Ryttov & Sannino 07

All Orders Beta Function

SU(N) Phase Diagram

Dietrich & Sannino 07

Sannino & Tuominen 04

Ryttov & Sannino 07

γm = 1 γm = 2

Fund

2A

2S

Adj

Ladder

Ryttov & Sannino 07

All Orders Beta Function

SU(N) Phase Diagram

Dietrich & Sannino 07

Sannino & Tuominen 04

Ryttov & Sannino 07

γm = 1 γm = 2

Fund

2A

2S

Adj

Ladder

Ryttov & Sannino 07

All Orders Beta Function

SU(N) Phase Diagram

Dietrich & Sannino 07

Sannino & Tuominen 04

Ryttov & Sannino 07

γm = 1 γm = 2

Fund

2A

2S

Adj

Ladder

Ryttov & Sannino 07

All Orders Beta Function

SU(N) Phase Diagram

Dietrich & Sannino 07

Sannino & Tuominen 04

Ryttov & Sannino 07

γm = 1 γm = 2

3

2

1

Sannino Tuominen

Adjoint - SU(2)Minimal Walking Technicolor

Catterall, Sannino 07Catterall, Gidet, Sannino, Schneible 08Del Debbio, Patella, Pica 08Del Debbio et al.. 09Hietanen, Rummukainen, Tuominen 09Catterall, Giedt, Sannino Schneible 09Bursa, Del Debbio, Keegan, Pica, Pickup

Υ=2 Ryttov Sannino

Conformal Chiral Symmetry Breaks ?

SD

Dual Sannino

Ryttov Sannino

Υ=1

NDualf ≥ 8.25

4

3

2

1

Dual - Fund

Dual - 2-index

Sannino Tuominen

Sannino

Sannino

Sym - SU(3)

Shamir, Svetitsky, DeGrand 08 DeGrand 09

Ryttov SanninoΥ=1

Υ=2 Ryttov Sannino

Conformal Chiral Symmetry Breaks ?

SD

11

10

9

8

7

6

5

4

3

2

SU(2) Fundamental

RS = Ryttov-Sannino 2007

RSΥ=1

Υ=2 RS

ATW = Appelquist-Terning-Wijewardhana 97

ATW

ACS = Appelquist-Cohen-Schmaltz 99

ACS

Muraya, Nakamura, Nonaka 03Skullerud et al. 04Iwasaki et al. 04

Iwasaki et al. 04

Conformal Chiral Symmetry Breaks ?

99

F.S. 09

SU(3) Fundamental17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

RS = Ryttov-Sannino 2007

RSΥ=1

Υ=2 RS

Dual S = Sannino 2009

FSDual

FSDual

ATW = Appelquist-Terning-Wijewardhana 97

ATW

ACS = Appelquist-Cohen-Schmaltz 99

ACS

Sui 01Hasenfratz 09Fodor et al. 09

Yamada et al. 09

Heller 98Hasenfratz 09Fodor et al. 09

Hasenfratz 09Fodor et al. 09Appelquist, Fleming, Neil 07Deuzeman, Lombardo, Pallante 09Jin, Mawhinney 09

Iwasaki et al. 04

Appelquist, Fleming, Neil 07Deuzeman, Lombardo, Pallante 09Fodor et al. 09Jin, Mawhinney 09

Fodor et al. 09

Conformal Chiral Symmetry Breaks ?

Be imaginative!

5 6 7 8 9 100

5

10

15

20

N

Nf

A.F.B.F. & γ= 2 SD

SO(N)

F.S. 09All Orders Beta Function

SO(N) Phase Diagram

F.S. 09All Orders Beta Function2 3 4 5 60

5

10

15

20

25

30

35

N

N Wf

A.F. B.F. & γ= 2 SD

SP(2N)

Sp(N) Phase Diagram

Chiral Gauge Theories

What does chiral mean?

Cannot give a mass term in the Lagrangian

What does chiral mean?

Cannot give a mass term in the Lagrangian

Why study them?

What does chiral mean?

Cannot give a mass term in the Lagrangian

The Standard Model is chiral

Why study them?

What does chiral mean?

Cannot give a mass term in the Lagrangian

Grand Unifications

The Standard Model is chiral

Why study them?

What does chiral mean?

Cannot give a mass term in the Lagrangian

Grand Unifications

The Standard Model is chiral

Why study them?

Extended technicolor models

What does chiral mean?

Cannot give a mass term in the Lagrangian

Grand Unifications

The Standard Model is chiral

Why study them?

Extended technicolor models

First principle lattice computations are impossible

Bars - Yankielowicz (BY)

Bars - Yankielowicz (BY)

Generalized Georgi-Glashow (GGG)

F.S. 09

Chiral beta function conjecture

vector-like pairsp(ri)

γi

β0

mass anomalous dimension

one-loop coefficient

β�χ Fixed to agree with the 2-loop coefficient

Chiral Gauge Theories Phase Diagram

F.S. 09Chiral Beta Function

2 4 6 8 10

5

10

15

20

N

p

2 4 6 8 10

5

10

15

20

Np

BY GGG

ϒ=2ϒ=2

ϒ=1ϒ=1

ACS

ACS

Thermal: Appelquist, Cohen, Schmaltz, Shrock 99 Appelquist, Duan, F.S., 2000

Poppitz & Unsal have similar results with a different method

Duals

T1

UV

T2T1

T1

IR

UV

T2T1

T1

IR

UV

T2

Electric

T1

Magnetic

T1 Weak

Electric

Strong

Magnetic

T1 Weak

Electric

Strong

Magnetic

e g = 2 π n

QCD Dual F.S. 09

QCD Dual F.S. 09

Within the Conformal Window

QCD Dual

‘t Hooft Anomaly Conditions Respected

F.S. 09

Within the Conformal Window

QCD Dual

‘t Hooft Anomaly Conditions Respected

F.S. 09

Within the Conformal Window

Operator Matching

QCD Dual

‘t Hooft Anomaly Conditions Respected

F.S. 09

Within the Conformal Window

Operator Matching

Flavor Decoupling

QCD Dual

‘t Hooft Anomaly Conditions Respected

F.S. 09

Within the Conformal Window

Operator Matching

Flavor Decoupling

RS beta function

QCD Dual

‘t Hooft Anomaly Conditions Respected

F.S. 09

Within the Conformal Window

Operator Matching

Flavor Decoupling

RS beta function

Super QCD Seiberg 96

QCD Dual

‘t Hooft Anomaly Conditions Respected

F.S. 09

Within the Conformal Window

Operator Matching

Flavor Decoupling

RS beta function

Super QCD Seiberg 96

Previous attempt Terning 98

T1

T1

T2

‘t Hooft Anomaly Matching

UV (1)

SUL(Nf )

SUL(Nf )

SUL(Nf )

SUL(Nf )

SUL(Nf )

Operator Matching

Baryonic bound states

Magnetic Electric

Baryons

M ∼ QQ

A, S, ....

M

�c1....cX qc1 · · · qcX

A QCD Dual F.S. 09

Dual Theory

LDual = LKin

�q, q, A,BA, DA, A

�+ LM

Dual Theory

LM = Yqeq q M �q + YABA A MBA + YCBA C MBA + YCBS C MBS + YSBS S MBS ++ YBADA BA M DA + YBADS BA M DS + YBSDA BS M DA + YBSDS BS M DS + h.c.

LDual = LKin

�q, q, A,BA, DA, A

�+ LM

Dual Theory

LM = Yqeq q M �q + YABA A MBA + YCBA C MBA + YCBS C MBS + YSBS S MBS ++ YBADA BA M DA + YBADS BA M DS + YBSDA BS M DA + YBSDS BS M DS + h.c.

LDual = LKin

�q, q, A,BA, DA, A

�+ LM

B[�c1....cX qc1 · · · qcX ] = 3 (2Nf − 5) = #×B[�c1....c3Qc1 · · · Qc3 ]

Nf

N

Electric

Strong

Weak

Nf

N

Magnetic

Electric

Weak

StrongStrong

Weak

Magnetic Asymp. Freedom

β0 =113

(2Nf − 5N)− 23Nf ≥ 0

Magnetic Asymp. Freedom

β0 =113

(2Nf − 5N)− 23Nf ≥ 0

Nf ≥114

N

Magnetic Asymp. Freedom

β0 =113

(2Nf − 5N)− 23Nf ≥ 0

Nf ≥114

N NBFf |γ=2 ≥

114

N

Magnetic Asymp. Freedom

ElectricChiral Sym. Restored

• Duality can be tested on the Lattice

• Reduces the number of theories

• Use to compute nonperturbative quantities like the VV-AA correlator.

Duality summary

Low Energy Effective TheoryMany Models

(?)MSSM XLMS D

AdS/?Unparticle

Branes......

Technicolor

Low Energy Effective Theory Unifying in Model Space

(?)MSSM

XLMS D

AdS/?

Unparticle

Branes

Technicolor?

Low Energy Effective TheoryMinimal Superconformal TC

MSSM

Strings

AdS/CFT

Unparticle

Branes

Technicolor

MSCT

Conclusions

๏ DEWSB can naturally occur at the LHC

Conclusions

๏ DEWSB can naturally occur at the LHC

๏ Phase Diagram of strongly interacting theories

Conclusions

๏ DEWSB can naturally occur at the LHC

๏ Phase Diagram of strongly interacting theories

๏ Unification in theory space

Conclusions

๏ DEWSB can naturally occur at the LHC

๏ Phase Diagram of strongly interacting theories

๏ Unification in theory space

๏ DEWSB cosmology is exciting.. another time

Conclusions